1 Approximating functions, polynomial interpolation (Lagrange and Newton’s divided differences) formulas, error approximations

2 Introduction to Interpolation  Introduction  Interpolation Problem  Existence and Uniqueness  Linear and Quadratic Interpolation  Newton’s Divided Difference Method  Properties of Divided Differences

3 Introduction Interpolation was used for long time to provide an estimate of a tabulated function at values that are not available in the table. What is sin (0.15)? xsin(x) Using Linear Interpolation sin (0.15) ≈ True value (4 decimal digits) sin (0.15) =

4 The Interpolation Problem Given a set of n+1 points, Find an n th order polynomial that passes through all points, such that:

5 Example An experiment is used to determine the viscosity of water as a function of temperature. The following table is generated: Problem: Estimate the viscosity when the temperature is 8 degrees. Temperature (degree) Viscosity

6 Interpolation Problem Find a polynomial that fits the data points exactly. Linear Interpolation: V(T)= 1.73 − T V(8)=

7 Existence and Uniqueness Given a set of n+1 points: Assumption: are distinct Theorem: There is a unique polynomial f n (x) of order ≤ n such that:

8 Examples of Polynomial Interpolation Linear Interpolation  Given any two points, there is one polynomial of order ≤ 1 that passes through the two points. Quadratic Interpolation Given any three points there is one polynomial of order ≤ 2 that passes through the three points.

9 Linear Interpolation Given any two points, The line that interpolates the two points is: Example : Find a polynomial that interpolates (1,2) and (2,4).

10 Quadratic Interpolation  Given any three points:  The polynomial that interpolates the three points is :

11 General n th Order Interpolation Given any n+1 points: The polynomial that interpolates all points is:

12 Divided Differences

13 Divided Difference Table xF[ ]F[, ]F[,, ]F[,,,] x0x0 F[x0]F[x 0,x 1 ]F[x 0,x 1, x 2 ]F[x 0,x 1,x 2,x 3 ] x1x1 F[x 1 ]F[x 1,x 2 ]F[x 1,x 2,x 3 ] x2x2 F[x 2 ]F[x 2,x 3 ] x3x3 F[x 3 ]

14 Divided Difference Table f(x i ) xF[ ]F[, ]F[,, ] Entries of the divided difference table are obtained from the data table using simple operations.

15 Divided Difference Table f(x i ) xF[ ]F[, ]F[,, ] The first two column of the table are the data columns. Third column: First order differences. Fourth column: Second order differences.

16 Divided Difference Table xF[ ]F[, ]F[,, ]

17 Divided Difference Table xF[ ]F[, ]F[,, ]

18 Divided Difference Table xF[ ]F[, ]F[,, ]

19 Divided Difference Table xF[ ]F[, ]F[,, ] f 2 (x)= F[x 0 ]+F[x 0,x 1 ] (x-x 0 )+F[x 0,x 1,x 2 ] (x-x 0 )(x-x 1 )

20 Two Examples xy Obtain the interpolating polynomials for the two examples: xy What do you observe?

21 Two Examples xY xY Ordering the points should not affect the interpolating polynomial.

22 Properties of Divided Difference Ordering the points should not affect the divided difference:

23 Example  Find a polynomial to interpolate the data. xf(x)

24 Example xf(x)f[, ]f[,, ]f[,,, ]f[,,,, ]

25 Summary

26 Lagrange Interpolation

27 The Interpolation Problem Given a set of n+1 points: Find an n th order polynomial: that passes through all points, such that:

28 Lagrange Interpolation Problem: Given Find the polynomial of least order such that: Lagrange Interpolation Formula: ….

29 Lagrange Interpolation

30 Lagrange Interpolation Example x1/31/41 y27

31 Example Find a polynomial to interpolate: Both Newton’s interpolation method and Lagrange interpolation method must give the same answer. xy

32 Newton’s Interpolation Method 012-3/2 7/6-5/ /

33 Interpolating Polynomial

34 Interpolating Polynomial Using Lagrange Interpolation Method

35 Inverse Interpolation Error in Polynomial Interpolation

36 Inverse Interpolation …. One approach: Use polynomial interpolation to obtain f n (x) to interpolate the data then use Newton’s method to find a solution to x

37 Inverse Interpolation …. Inverse interpolation: 1. Exchange the roles of x and y. 2. Perform polynomial Interpolation on the new table. 3. Evaluate ….

38 Inverse Interpolation x y y x

39 Inverse Interpolation Question: What is the limitation of inverse interpolation? The original function has an inverse. y 1, y 2, …, y n must be distinct.

40 Inverse Interpolation Example x123 y

41 Errors in polynomial Interpolation  Polynomial interpolation may lead to large errors (especially for high order polynomials). BE CAREFUL  When an n th order interpolating polynomial is used, the error is related to the (n+1) th order derivative.

42 10 th Order Polynomial Interpolation

43 Errors in polynomial Interpolation Theorem

44 Example

45 Summary  The interpolating polynomial is unique.  Different methods can be used to obtain it. Newton’s divided difference Lagrange interpolation Others  Polynomial interpolation can be sensitive to data.  BE CAREFUL when high order polynomials are used.